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Is there a relativity-compatible thermodynamics?

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I am just wondering that laws in thermodynamics are not Lorentz invariant, it only involves the $T^{00}$ component. Tolman gave a formalism in his book. For example, the first law is replaced by the conservation of energy-momentum tensor. But what will be the physical meaning of entropy, heat and temperature in the setting of relativity? What should an invariant Stephan-Boltzmann's law for radiation take shape? And what should be the distribution function?

I am not seeking "mathematical" answers. Wick rotation, if just a trick, can not satisfy my question. I hope that there should be some deep reason of the relation between statistical mechanics and field theory. In curved spacetime, effects like particle production seems very strange to me, since they originate from the ambiguity of vacuum state which reflects the defects of the formalism. The understanding of relativistic thermodynamics should help us understand the high energy astrophysical phenomena like GRB and cosmic rays.

I belive that the trick to get a relativistic version of entropy, temperature and etc is to define things in the proper reference frame, since you can extend it latter to other references keeping lorentz covariance. At least this is one way to arrive at relativistic hydrodynamics. About the QFT and GR: I don't have any idea

I know you don't want to hear it but I think Wick Rotation is really the right thing. The connection between the classical world and the quantum one is the path integral, through the action. This connects the partition function to the generating function and you're done, baring the Wick rotation. Now in curved space I don't have an answer, because even the definition of the Hilbert space (a la Hawking) is tricky. But then, QFT doesn't play well with GR, so we wouldn't expect the connection to extend that far anyway.

Thermodynamics is compatible with special relativity. Tolman's book was concerned about General Relativity, and he discovered interesting effects you can do with heat engines in gravitational fields. For example, he gave a theoretical heat engine which operates at the Carnot limit without being infinitesimally slow, by lifting and lowering things in a gravitational field to change temperature (if I remember correctly). This was considered impossible before. I don't remember the exact setup, so I can't vouch for what he did, I read the book a long time ago.

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I don't know a definitive answer to your (really good) question, but here is a quote from an old textbook I have by Christian Moller ("The Theory of Relativity"):

Shortly after the advent of the relativity theory, Planck, Hassenoerl,
Einstein and others advanced separately a formulation of the
thermodynamical laws in accordance with the special principle of
relativity. This treatment was adopted unchanged including the first
edition of this monograph. However it was shown by Ott and indepently
by Arzelies, that the old formulation was not quite satisfactory, in
particular because generalized forces were used instead of the true
mechanical forces in the description of thermodynamical processes.

The papers of Ott and Arzelies gave rise to many controversial
discussions in the literature and at the present there is no
generally accepted description of relativistic thermodynamics.

So at least at the time that was written it was unresolved. I'd be interested if there are any more recent updates.

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